Question

N 133 (5х-2)/3-(3-х)/2>1 N 142 а)х^2-3,2x<0; b)(3x-2)^2-4x(2x-3)>=0

Answer 1

A)
x(x-3.2)<0
нули:
x=0 x=3.2
Ответ (0;3,2)

Answer 2

Task 1: Solve the inequality (5x-2)/3 - (3-x)/2 > 1

To solve the given inequality, let's simplify it step by step:

Step 1: Multiply both sides of the inequality by 6 to eliminate the denominators: 6 * [(5x-2)/3 - (3-x)/2] > 6 * 1

Simplifying the left side: 2(5x-2) - 3(3-x) > 6

Step 2: Expand and simplify the expression: 10x - 4 - 9 + 3x > 6 13x - 13 > 6

Step 3: Add 13 to both sides to isolate the variable: 13x - 13 + 13 > 6 + 13 13x > 19

Step 4: Divide both sides by 13 to solve for x: (13x)/13 > 19/13 x > 19/13

Therefore, the solution to the inequality (5x-2)/3 - (3-x)/2 > 1 is x > 19/13.

Task 2:

a) Solve the inequality х^2 - 3.2x < 0

To solve the inequality, let's factor the quadratic expression:

Step 1: Factor out x from the expression: x(x - 3.2) < 0

Step 2: Set each factor equal to zero and solve for x: x = 0 x - 3.2 = 0 x = 3.2

Step 3: Plot the critical points on a number line: -infinity 0 3.2 infinity

Step 4: Test a value from each interval to determine the sign of the expression: Test x = -1: (-1)(-1 - 3.2) < 0 => 1.2 < 0 (False) Test x = 1: (1)(1 - 3.2) < 0 => -2.2 < 0 (True) Test x = 4: (4)(4 - 3.2) < 0 => 3.2 < 0 (False)

Step 5: Based on the test results, we can conclude that the expression is negative when x is between 0 and 3.2, inclusive. Therefore, the solution to the inequality х^2 - 3.2x < 0 is 0 ≤ x ≤ 3.2.

b) Solve the inequality (3x-2)^2 - 4x(2x-3) ≥ 0

To solve the inequality, let's simplify it step by step:

Step 1: Expand and simplify the expression: (3x-2)^2 - 4x(2x-3) ≥ 0 (9x^2 - 12x + 4) - (8x^2 - 24x) ≥ 0 9x^2 - 12x + 4 - 8x^2 + 24x ≥ 0 x^2 + 12x + 4 ≥ 0

Step 2: Plot the critical points on a number line: -infinity

Step 3: Since the leading coefficient of the quadratic expression is positive, the parabola opens upward. Therefore, the expression is non-negative for all values of x.

Step 4: The solution to the inequality (3x-2)^2 - 4x(2x-3) ≥ 0 is all real numbers.

Please let me know if anything is unclear or if you need further assistance!